Canonical and micro-canonical typical entanglement of continuous variable systems

Abstract

We present a framework, compliant with the general canonical principle of statistical mechanics, to define measures on the set of pure Gaussian states of continuous variable systems. Within such a framework, we define two specific measures, referred to as `micro-canonical' and `canonical', and apply them to study systematically the statistical properties of the bipartite entanglement of n-mode pure Gaussian states (as quantified by the entropy of a subsystem). We rigorously prove the "concentration of measure" around a finite average, occurring for the entanglement in the thermodynamical limit in both the canonical and the micro-canonical approach. For finite n, we determine analytically the average and standard deviation of the entanglement (as quantified by the reduced purity) between one mode and all the other modes. Furthermore, we numerically investigate more general situations, clearly showing that the onset of the concentration of measure already occurs at relatively small n.